Friday, September 2, 2011

On Left/Right Symmetry, the Platonic Ideal, and Godel's Theorem

Plato, who lived in Athens from (429 to 347 BC), believed that there exists a realm of perfect, ideal shapes. He believed that the shapes we see, such as the circles we try to draw on pieces of paper, are but imperfect Earthly forms of a perfect, eternal ideal that we can't see.
The idea that everything is but an imperfect example of a smaller set of pure objects might be fine and dandy for certain mathematical objects, such as the circle or the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron), but the idea of a perfect object breaks down for living creatures. Unfortunately, the philosophy of idealism as developed by Plato and Socrates hindered the eventual development of the theory of evolution because many people believed that each species was related only to its eternal ideal form, and not to other species. I will point out in this post the error in thinking that there could be such a thing as a perfect cat or a perfect elephant or a perfect homo sapiens.

Throughout the history of life, there have been "either/or" symmetry-breaking moments. One of the first examples was when life forms used and grew only left-handed amino acids, instead of right-handed amino acids. (For more information on the difference between left and right handed molecules follow this link.) I think it's important to stop and ask: why did life develop the ability to use only left-handed amino-acids? (Note that there are a few bacteria that use right handed amino-acids, but the overwhelming majority use left-handed amino-acids.) It turns out that this is still an open question because the evidence scientists have collected to answer the question doesn't directly confirm any one theory.

Here's the main evidence collected so far to try to answer this question.
1) The Miller-Urey experiments produced both left-handed and right-handed amino acids in equal amounts. This suggests that the left vs. right choice was a fluke, i.e. there was a 50%-50% chance that one type would dominate, and perhaps there was a 50% chance that we would have used only right-handed molecules instead.
2) Meteors seem to produce 7%-9% more left-handed than right-handed amino-acids. This suggests that perhaps left-handed amino-acids survive better than right-handed one, but even a 7%-9% initial difference means that there's still a large chance that life could have developed the use of right-handed molecules instead. Even so, we are left with multiple questions: a) why aren't ~40% of the bacteria able to right handed amino acids? b) why can't bacteria use both left and right handed amino acids...and both left and right handed sugars? c) why are there more left-handed than right handed amino acids on the meteors? d) What is the initial cause of the asymmetry and is the cause of the asymmetry in the meteors the same as the asymmetry on Earth.
3) We know of only one force in nature than can distinguish between left and right, and that's the weak nuclear force. This force does not conserve parity, i.e. the weak nuclear force can be used to distinguish left and right handed particles. For example, only left-handed neutrinos or right-handed anti-neutrinos can participate in reactions via the weak nuclear force. No other force is that unique (not gravity, not electromagnetism, and not the strong nuclear force.) But what does the weak nuclear force have to do with anything here on Earth? Or on the meteors mentioned above? Does the weak nuclear force have anything to do with why life developed the capability to use a certain chiral form of amino-acids, DNA and sugars, but not the opposite chiral form? I can't think of how this would be possible because life does not appear to be dependent on the weak nuclear force...only gravity to hold us on Earth, E&M to hold protons and electrons together, and the strong nuclear force to hold the nucleus together. I can't think of any way in which the weak nuclear force could effect the structure of large molecules like DNA, amino acids and sugars. (But if the weak nuclear force doesn't effect life, how does a human zygote nearly always develop such that the heart of the fetus is on the left side of the body? How does the zygote distinguish between left and right if gravity, E&M and the strong nuclear force can't distinguish between left and right? Does it rely on the handedness of DNA and amino acids?)

So, this leaves me with even more questions: Was the eventual dominance of left-versus-right due to initial offsets? (Perhaps the universe itself is asymmetric) Or more likely, was it just dumb luck? And by dumb luck, I mean: was it the particular chirality (i.e the handedness) of the first RNA or DNA that caused the exponential growth of a certain type of amino-acid via self-replication.

But there were so many left-right choices in the history of human development, that I'm pretty much forced to conclude that it was dumb luck, i.e. we could have developed the use of right handed amino acids. Other left-right 'choices' include: a) which side our heart develops b) which side of our brain develops certain storage locations c) why 90% of the population is right handed, and d) why the left testicle for men hangs lower than the right on average

The point I'm trying to make here is that there have been so many left-right (or up-down) choices in the history of life that there's no way to think that the left-left-up-right-down-right-up (etc...) that makes current humans was due to anything but dumb luck.

But how does this seemingly breaking of left-right symmetry stand up when compared to the Rosen-Curie Principle, which states that the symmetry of the effect is greater than the symmetry of the cause. If life can only become more symmetric, then how can we account for the dominance of left-handed amino-acids? Why does only one type of amino acid grow exponentially? In order to not violate the Rosen-Curie Principle, we must accept that the universe was not perfectly mirror symmetric to start, because if it were perfectly mirror symmetric to start, then there would be no way to have only left-handed amino acids grow exponentially. This suggests that the initial conditions of the universe were not left-right symmetric. It is very likely that the universe started in a state of low symmetry.

And now, I'd like to show how Gödel's incompleteness theorem effectively destroys Plato's idea that there could be such a thing as an ideal or universal 'tree' or 'bird.'
Gödel showed that, for sufficiently strong and well-formed set of axioms like the set of axioms for arithmetic, there are statements (that we'll call 'G') that can't be prove to be true or false. And further, if we now add a new axiom to the set of axioms which states "G is false" or an axiom which states "G is true", there will be new statements (that we'll call F & H) that can't be proved to be true or false in the new sets of axioms. We could continue...in a set of axioms with "G is true" and "H is true" there will be a statement 'J' which can not be proven to be true or false, and this means that we can now add either "J is true" or "J is false" and make a new set of axioms...which will still be incomplete...i.e. there will still be statements out there which can't be proven to me true or false.

And now, I'd like to make an analogy with life.
It's as if life is a program for generating new programs. When it reaches an axiom (i.e. food) that it can't determine whether it fits into its existing program, it has to make a choice and either include the axiom as true (i.e. left-handed food) or false (i.e. right-handed food.) Once it makes the decision, then all future programs make statements and new programs assuming that the axiom is true (or false.) There's generally no going back. Then, later on, it runs into a new axiom. Will it accept the axiom as true (create left-hand amino-acids) or false (create right-handed amino-acids)? There's no right answer to the question: should we accept this axiom as true or false? It's like we're little Gödel-esque programs constantly trying to calculate the truth or falseness of certain statements, and when we come upon a statement G that we can't prove is either true or false, we simple make a choice and incorporate either "G is true" or "G is false" into our set of programs.
The analogy here is between "truth & false" and "right and left handed." (This is not intended to be a perfect analogy; it's only intended to convey the sense that there seem to be a lot of times in the history of life in which there were completely arbitrary choices made between left-handed and right-handed chemical, just as the choice of either adding the statement "G is true" or adding the statement "G is false" is completely arbitrary. There was no way to prove that G was true or false, so you just pick one and move on.)

In the case of mathematical sets of axioms, you have to pick "G is true" or "G is false." You can't pick both. (Or else you don't have a consistent set of axioms.) But is the same true for life? Could bacteria have developed the capability to harness both left and right handed food molecules? Perhaps it did initially. But if so, why did that ambidextrous form of life not out grow and out multiple the only-left or only-right handed lifeforms? You'd think that a bacteria that could use both left and right handed molecules could grow faster than a bacteria that could only use one handedness of molecules. What's the advantage to just-left or just-right molecules?

My guess is that trying to use both types ends up creating DNA molecules that are both right and left handed, and since left-handed DNA will not match up with right handed DNA, there might be a problem with trying to use both right and left handed amino-acids. The use of certain sugars probably also stems from the fact that sugars are the backbone of DNA. You can't afford to have waste your time building a strand of DNA and then have a wrong-handed sugar molecule show up and ruin the spiral.

Life 'choses' one handedness over the other, and this happened over and over and over again throughout the development of life.

And so now we see the problem with Plato's idea that there is a world of ideal forms. There is no such thing as an ideal form for living creatures because an ideal form can't be arbitrary. If life is made up of sets of arbitrary choices between left and right, then there is no such thing as an ideal life form because how could an 'ideal life form' be arbitrary. For example, would the ideal 'tree' use left or right handed amino-acids? Would that same 'ideal tree' spiral clockwise or counterclockwise as it grew?

Each living creature is like those sets of axioms that Gödel thought about...it is made up a series of completely arbitrary choices between left and right. Living creatures are irreducibly complex. There are just too many arbitrary left-right choices, and each of those left-right choices means that there is no ideal species. For example, would the ideal human have their heart on the left or the right? Would it be in the middle?

As the physicist Roger Penrose wrote, "If the axiom of choice [i.e. "G is true" or "G is false"] is a mere matter of opinion or of arbitrary decision, then the Platonic world of absolute mathematical forms contains neither the axiom of choice nor its negation." Penrose is suggesting here that mathematics (such as algebra) can't exist in some Platonic ideal world becuase the choice of axioms are completely arbitrary. We can continue with Penrose's thinking in "The Road to Reality"... There is no such thing as a generally true mathematic statement. There's only statements that are true if you assume axiom x,y,z,... are true. But if you assume a different set of starting axoims, then the statement might not be true. For example, is the statement "The sum of all angle for a triangle equal to 180 degrees" a true statement? It's true if you assume certain axioms (such as for planar geometry), but it's not a true statement if you assume different axioms (such a hyperbolic geometry.) So,we can't even say that "The sum of all angle for a triangle equal to 180 degrees" is always a true statement.
Is there somebody to tell us which set of axioms is true? The answer is no. Can God tell us whether the statements of mathematics are always true? The answer, once again, is no.

The laws of mathematics are not handed down to us from God or some Platonic realm of ideal equations. And further, Plato's ideal world can not hold life forms because it can't contain such impure 'left-handed' creatures, such as we are. While it's tough to ponder the abundance of arbitrary choices between left and right, I think that it's an interest topic to discuss between there are a lot of open question:
1) Why did life develop the capability to use only left handed amino acids? (Nearly only left)
2) How does a zygote distinguish between left from right if the three forces that effect life (gravity, E&M and SN) are left-right symmetric? Does the zygote use the chirality of molecules to tell the difference? In which case, we go back to question #1.

What's so ironic is that Gödel considered himself to be a Platonist, i.e. he believed to some extent that mathematical truths exist outside of this world. But his first and second incompleteness theorems, as well as his work on arbitrariness of the axioms of set theory, are the last nail in the coffin of Platonism because his research proves the arbitrariness of the axioms that we use in mathematics, and this is very similar to the arbitrariness of many of the left-right 'choices' that life has made throughout its exist on Earth.

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Three Worlds, Three Mysteries

The Goal of this Blog

My goal is to communicate how life can expand and grow, both on this planet and on others. To grow, we need to obtain a large rate of return on investment from our power plants, so a main focus of this blog is on the economics of electricity generation and vehicle transportation.To summarize, the goal of life is to expand. Life requires mechanical or electro-chemical work to survive, and to grow, it requires a large, positive rate of return on work invested.

In other words, the purpose of a power plant is to make more power plants, and as quickly as possible.

After a series of posts on the topic of energy policy and economics, I thought that it'd be a good time to take a break and delve back i...

Good quotes

"The [engineer] should be equipped with knowledge of many branches of study and varied kinds of learning, for it is by his judgement that all work done by the other arts is put to test. This knowledge is the child of practice and theory."

Marcus Vitruvius Pollio, De Architectura, (~15 BC)

"The fact that the Standard Model of Physics has just enough complexity to be able to accommodate CP violation does not shed any light on the true nature of this phenomenon, and we feel the conclusion of J. Cronin's 1980 Nobel speech still stands: 'We must continue to seek the origin of the CP symmetry violation by all means at our disposal. [...] We are hopeful, then, that at some eposh, perhaps distant, this cryptic message from nature will be deciphered.' (Cronin, 1981) --Marco Sozzi, 2008, from the Coda of Discrete Symmetries and CP Violation

"Knowledge is power."

"Lastly, I would address one general admonition to all; that they consider what are the true ends of knowledge, and that they seek it not either for pleasure of the mind, or for contention, or for superiority to others, or for profit, or fame, or power, or any of these inferior things; but for the benefit and use of Life; and that they perfect and govern it in charity."— Francis Bacon

"Dare to be an optimist."

—Matt Ridley

"Stretch the range of human powers...Give us new metaphors with which to puzzle out our mysteries...Give us pride and higher aspirations...Turn our trash into treasure...Give us goals and meaning...Give us new tools with which we can connect...Validate us in our moments of confusion...Give us new rituals to make sense of our day...Give us new levels of reality...Give us your soul and bare your emotions...Give us new tools of understanding...Turn luxuries into everyday commodities...Warn us of our failings, of our conplacency, or our alternatives and of our dangers...Help us serve a purpose higher than ourselves."

—Howard Bloom, The Genius of the Beast

“Progress is possible only when people believe in the possibilities of growth and change. Races or tribes die out not just when they are conquered and suppressed, but when they accept their defeated condition, become despairing, and lose their excitement about the future.”—Norman Cousins

"If I had to choose a religion, the sun as the universal giver of life would be my god."— Napoleon Bonaparte

"Knowledge is of no value unless you put it into practice."— Anton Chekhov

"There is only one good, knowledge, and one evil, ignorance."

"Let no one ignorant of Mathematics enter [the Academy]."

— Plato/Socrates

"Perhaps the only goal on earth to which mankind is striving lies in this incessant process of attaining, in other words, in life itself..."

— Narrator of Dostoevsky's Notes from the Underground

"The physical holds no power over destiny."

—Norman Paperman

"Nature is not solely physical, even though everything in it must depend on the physical. Nature includes not only physical entities but complex material organizations, mathematical lawfulness, possibilities, life, need, behavior, intelligence, purpose, societies, minds, meanings, signs, and knowledge."

— Lawrence Cahoone, "The Orders of Nature"

"All men by nature desire knowledge."— Aristotle

"Worrying is praying for something that you don't want. So stop worrying!"

—Bhagavan Das

"Keep your head above the water and bet on the growth of your country."— Henry Flagler of Standard Oil

"Society will develop a new kind of servitude which covers the surface of society with a network of complicated rules, through which the most original minds & and the most energetic of characters cannot penetrate. It does not tyrannize, but it compresses, enervates, extinguishes & stupefies a people until each nation is reduced to nothing better than a flock of timid & industrious animals, of which the government is the shepherd."

"Socrates: 'Now, that which imparts truth to the known and the power of knowing to the knower is what I would have you term the idea of good, and this you will deem to be the cause of science, and of truth in so far as the latter becomes the subject of knowledge; beautiful too, as are both truth and knowledge, you will be right in esteeming this other nature as more beautiful than either; and, as in the previous instance, light and sight may be truly said to be like the sun, and yet not the sun, so in this other sphere, science and truth may be deemed to be like the good, but not the good; the good has a place of honor yet higher.'Glaucon: 'What a wonder of beauty that must be, which is the author of science and truth, and yet surpasses them in beauty; for you surely cannot mean to say that pleasure is the good?'

Socrates: 'God forbid, but I may ask you to consider the image in another point of view.'

Glaucon: 'In what point of view?'

Socrates: 'You would say, wouldn't you not, that the sun is not only the author of visibility in all visible things, but of generation and nourishment and growth, though he himself is not generation?'

Glaucon: 'Certainly.'

Socrates: 'In like manner, the good may be said to be not only the author of knowledge to all things known, but of their being and essence, and yet the good is not essence, but far exceeds essence in dignity and power.' "